Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

Abstract

We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not have a symmetric extension. For the specific case of 2-qutrit states, we use numerical evidence to conjecture that the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence, the violation of the CGLMP inequality by any 2-qutrit state could be a sufficient condition for the non-existence of its symmetric extension.

Extendibility of Bosonic Gaussian States

“…For fixed k, the extendibility problem can be formulated as a semidefinite program (SDP), making it efficiently solvable for low-dimensional systems A and B [5,6]. Analytic conditions for k-extendibility in finite-dimensional systems are known only for particular values of k and/or for special classes of states [24,[37][38][39][40].…”

“…The nested sets of k-extendible states can thus be used to approximate the set of separable states, which has resulted in work on quantum de Finetti theorems [8][9][10][11][12][13][14] and other studies of entanglement [15,16]. Extendibility also arises in the contexts of security of quantum key distribution [17][18][19], capacities of quantum channels [20][21][22], Bell's inequalities [23,24], and other information-theoretic scenarios [25,26]. More broadly, the extendibility problem is a special case of the QMAcomplete quantum marginal problem [27][28][29][30][31][32][33], which has been referred to in quantum chemistry as the N-representability problem [34][35][36].…”